特征根为单实根的连续系统的能观丰富性计算

     在博文“线性连续系统的能观丰富性”(http://blog.sciencenet.cn/blog-3343777-1067700.html)中,定义了能观丰富性并给出了其计算式如下

$v_{o,T}=\mathrm{Vol}(R_{o,T})=\left|W_{o,T}\right|^{-1}\mathrm{Vol}(\widetilde{R}_{o,T})$

       当系统为SISO时，系统矩阵为 $A$ 为对角阵且特征值 $\lambda_{i}(i=1,2,\cdots,n)$ 都为单实根, $C=[c_{1},c_{2},\cdots,c_{n}]$ ,则有

$W_{o,T}=\left[\begin{array}{ccc} c_{1}^{2}\frac{e^{2\lambda_{1}T}-1}{2\lambda_{1}} & \cdots & c_{1}c_{n}\frac{e^{(\lambda_{1}+\lambda_{n})T}-1}{\lambda_{1}+\lambda_{n}}\\ \vdots & \ddots & \vdots\\ c_{1}c_{n}\frac{e^{(\lambda_{1}+\lambda_{n})T}-1}{\lambda_{1}+\lambda_{n}} & \cdots & c_{n}^{2}\frac{e^{2\lambda_{n}T}-1}{2\lambda_{n}} \end{array}\right]$

当 $\lambda_{i}\in(-\infty,0](i=1,2,\cdots,n)$ ，有

$\widehat{W}=\lim_{N\rightarrow\infty}W_{o,N}=-\left[\begin{array}{cccc} \frac{c_{1}^{2}}{2\lambda_{1}} & \frac{c_{1}c_{2}}{\lambda_{1}+\lambda_{2}} & \cdots & \frac{c_{1}c_{n}}{\lambda_{1}+\lambda_{n}}\\ \frac{c_{1}c_{2}}{\lambda_{1}+\lambda_{2}} & \frac{c_{2}^{2}}{2\lambda_{2}} & \cdots & \frac{c_{2}c_{n}}{\lambda_{2}+\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{c_{1}c_{n}}{\lambda_{1}+\lambda_{n}} & \frac{c_{2}c_{n}}{\lambda_{2}+\lambda_{n}} & \cdots & \frac{c_{n}^{2}}{2\lambda_{n}} \end{array}\right]$

      对 $\widehat{P}$ 的行列式值的值，可证明为

$\det\left(\widehat{W}\right)=G\left[\prod_{1\leq j_{1}

其中 $G$ 为关于特征值 $\lambda_{i}(i=1,2,\cdots,n)$ 的待定函数。此时

        $\lim_{N\rightarrow\infty}v_{o,N}=\frac{1}{\det\left(\widehat{P}\right)}\left|\left(\prod_{1\leq j_{1}

                  $=\frac{1}{G}\left|\left(\prod_{1\leq j_{1}